Properties of Geometric Progression: Definitions, Formulas, and Examples

Properties of Geometric Progression: Progression is the sequence of numbers in which numbers are related to each other with some common relation. The series of numbers obtained by adding or subtracting a fixed number is said to be in arithmetic progression. At the same time, the series of numbers obtained by multiplying or dividing with a constant number is said to be in geometric progression.

The numbers in the geometric progression are associated with a lot of properties. One of the basic properties is each term of the geometric progression is obtained by dividing or multiplying with the constant number except the first term. In this article, let us discuss more properties of the geometric progression and the concepts and formulas in detail.

Geometric Progression

Progression is the series of numbers that are related by a common relation. The series of numbers in which each number is obtained by dividing or multiplying the previous term by a constant number except the first term is called the geometric series or geometric progression.

Learn the Concepts of Geometric Progression

Thus, the geometric progression is the series of numbers related to each other with some common relation, obtained by multiplication or division. The common number used to multiply or divide each term of the geometric progression is called the common ratio. In geometric progression, the common ratio may be any positive or negative real number. The geometric progression is generally denoted as G.P.

The daily-life examples of geometric progressions are

1. Calculating the interest earned by the bank
2. Population growth

Formulas in Geometric Progression

Geometric progression is a series of numbers in which each term is obtained by multiplying the previous term by a fixed number(common ratio). The geometric progressions are generally written as \(a, a r, a r^{2}, a r^{3}, \ldots \ldots\)

Here, each term is obtained by multiplying the previous term by the common ratio \((r)\) except the first term \((a.)\)

Here, 

1. \(a-\) First-term
2. \(r-\) Common Ratio

Also, the common ratio is obtained by dividing the previous term by the common ratio by excluding the first term.

The formula used to calculate the general term or the \({n^{{\rm{th}}}}\) term or the last term of the geometric progression is calculated by using the formula given below:

Sum of Terms of Geometric Progression

The sum of the geometric series formula is used to find the total of all the terms of the given geometrical series. There are two types of geometric progressions, namely infinite or infinite series. So, there are different formulas to calculate the sum of series, which are given below:

Sum of Finite Terms of Geometric Progression

Finite geometric progression is the series of numbers, which has finite numbers. In the finite series, the last term is defined.

The sum of terms of a geometric progression is given by

\(S_{n}=\frac{a\left(r^{n}-1\right)}{r-1}=\) When \(r>1\) and \(S_{n}=\frac{a\left(1-r^{n}\right)}{1-r}\), when \(r<1\)

Sum of Infinite Geometric Series

In the geometric progression \(a, a r, a r^{2}, a r^{3}, \ldots \ldots \infty\)

The sum of terms of the above geometric series is found by using the simple formula, which is given below:

\({S_\infty } = \frac{a}{{1 – r}}; – 1 < r < 1\)

Properties of Geometric Progression

Thus, the geometric progression is the series of numbers related to each other with some common relation, obtained by multiplication or division. The common number that is used to multiply or divide each term of the geometric progression is called the common ratio.

Property-1: When we multiply each term of the geometric series with the non-zero constant number, the new series also forms geometric series with the same common ratio.

Let \(a_{1}, a_{2}, a_{3}, a_{4}, \ldots \ldots a_{n}\) an be the geometric progression with the common ratio \(r=\frac{a_{n+1}}{a_{n}}\)

If we multiply each term of the geometric progression: \(k a_{1}, k a_{2}, k a_{3}, k a_{4}, \ldots \ldots k a_{n}\) then the common ratio is given by \(r=\frac{k a_{n+1}}{k a_{n}}=\frac{a_{n+1}}{a_{n}}\)

Property-2: When we divide each term of the geometric series with the non-zero constant number, the new series also forms geometric series with the same common ratio.

Let \(a_{1}, a_{2}, a_{3}, a_{4}, \ldots \ldots . a_{n}\) an be the geometric progression with the common ratio \(r=\frac{a_{n+1}}{a_{n}}\)

If we divide each term of the geometric progression: \(\frac{a_{1}}{k}, \frac{a_{2}}{k}, \frac{a_{3}}{k}, \frac{a_{4}}{k}, \ldots \ldots \frac{a_{n}}{k}\), then the common ratio is given by \(r=\frac{\frac{a_{n+1}}{k}}{a_{\frac{n}{k}}}=\frac{a_{n+1}}{a_{n}} .\)

Property-3: The series of reciprocal terms of a geometric progression also forms a geometric series with the common ratio equal to the reciprocal of the common ratio of the given series.

Let \(a_{1}, a_{2}, a_{3}, a_{4}, \ldots \ldots . a_{n}\) be the geometric progression with the common ratio \(r=\frac{a_{n+1}}{a_{n}}\)

Then, the reciprocal of the terms \(\frac{1}{a_{1}}, \frac{1}{a_{2}}, \frac{1}{a_{3}}, \frac{1}{a_{4}}, \ldots \ldots \frac{1}{a_{n}}\) also forms a geometric progression with a common ratio: \(r=\frac{1}{\frac{a_{n+1}}{a_{n}}}\)

Property-4: The series of numbers, in which each term of the geometric progression raised to the same power (say \(n)\), also form a geometric progression with the common ratio is \(r^{n}\).

Let \(a_{1}, a_{2}, a_{3}, a_{4}, \ldots \ldots . a_{n}\) be the geometric progression with the common ratio \(r=\frac{a_{n+1}}{a_{n}}\).

Then, the series of numbers \(a_{1}^{n}, a_{2}^{n}, a_{3}^{n}, a_{4}^{n}, \ldots \ldots a_{n}^{n}\) also, a geometric series with a common ratio is \(\left(\frac{a_{n+1}}{a_{n}}\right)^{n}\).

Property-5: In a geometric progression, the product of the terms that are equidistant from the beginning and at the end of the series is always equal to the product of the first term and the last term of the geometric series.

Property-6: When the terms of a geometric progression are selected at equal intervals, then the new series is also geometric.

Property-7: The logarithm of non-zero negative numbers of each term in a geometric progression forms the series of the numbers in an arithmetic progression.

Let \(a_{1}, a_{2}, a_{3}, a_{4}, \ldots \ldots . a_{n}\) be the series of numbers in geometric progression, then \( \log a_{1}, \log a_{2}, \log a_{3}, \log a_{4}, \ldots \ldots \log a_{n}\)  are in arithmetic progression.

It’s converse is also true, such that if \(a_{1}, a_{2}, a_{3}, a_{4}, \ldots \ldots . a_{n}\) be the series of numbers in arithmetic progression, then \(\log \log a_{1}, \log \log a_{2}, \log \log a_{3}, \log \log a_{4}, \ldots \ldots \log \log a_{n}\)  are in geometric progression.

Property-8: The three non-zero numbers \(a, b, c\) are said to be in geometric progression, if and only if \(b^{2}=a c\)

Solved Examples – Properties of Geometric Progression:

Q.1. If the non-zero numbers 2,4,8,16,32, …. are in geometric progression, check the series of numbers, in which each number is obtained by multiplying by 2 are in geometric progression or not? If they are in geometric progression, find the common ratio.
Ans: Given \(2, 4, 8, 16, 32, ….\) are in geometric progression. Now the series of numbers is obtained by multiplying each term of the series by \(2\) are 
\(2×2, 2×4, 2×8, 2×16, 2×32, …..\)
\(4, 8, 16, 32, 64, …….\)
We know that the ratio of each term to the previous term is the same except the first term, then the series is said to be in geometric progression.
Now, the ratio second term to the first term \(=\frac{8}{4}=2\) similarly the ratio third term to the second term \(=\frac{16}{8}=2 .\)
So, the series of numbers that are obtained by multiplying each term of the given series by \(2\) are in geometric progression with a common ratio \(2.\)

Q.2. Check if the square of each number given in the geometric progression: 3,9,27,81, … is also in a geometric progression.
Ans: The given geometric progression is \(3, 9, 27, 81, ….\)
The square of each term of the given series is \(3^{2}, 9^{2}, 27^{2}, 81^{2}, \ldots \ldots\)
The ratio of each term to the previous term except the first term is \(\frac{9^{2}}{3^{2}}=\frac{27^{2}}{9^{2}}=\frac{81^{2}}{27^{2}}=9\) is the same.
So, the series of the numbers obtained by squaring each term of the given geometric progression is also in geometric progression.

Q.3. Check if the three non-zero numbers 4,6,9 are in geometric progression or not.
Ans: Given numbers are \(4, 6, 9\)
We know that three numbers \(a, b, c\) are said to be in geometric progression if and only if \(b^{2}=a c\)
Here also, \(6^{2}=36\) and \(4×9=36\)
So, the numbers \(4, 6, 9\) are in geometric progression.

Q.4. If we divide each term of the number in the geometric series: 3,9,27,81… by 3, find out if the new series of numbers are in geometric progression or not.
Ans: Given geometric series : \(3, 9, 27, 81,….\)
Divide each term of the given series by \(3.\)
The new series obtained is \(1, 3, 9, 27, …..\)
The ratio of the term to the previous term except the first term is \(\frac{3}{1}=\frac{2}{3}=\frac{27}{9}=3\) the same.
Hence, the new series obtained is in geometric progression.

Q.5. If the series of the numbers 2,4,8,16, ….. are in the geometric progression, then check ifthe reciprocal of the numbers of the series are in geometric progression or not.
Ans: Given geometric progression: \(2, 4, 8, 16, ….\)
The reciprocal of the numbers in the given series is \(\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \ldots \ldots\)
Here, the ratio of the terms: \(\frac{\frac{1}{4}}{2}=\frac{\frac{1}{8}}{4}=\frac{\frac{1}{16}}{\frac{1}{8}}=\frac{1}{2}\) is the same.
So, the series of the reciprocal of the numbers in the given series is in geometric progression.

Summary

In this article, we have studied the definition of geometric progression. This article also tells the general form, first term and the common ratio of the geometric progression. We have studied the formulas of common ratio, general term of the series, and the sum of the term in the geometric progression of finite and infinite series.

This article gives the various properties of the geometric progression with examples. This article studied the solved examples that help us to understand the properties of geometric progression easily.

General Term of a Geometric Progression

Frequently Asked Questions

Q.1. What is geometric progression?
Ans: Geometric progression is a special kind of series in which each term of the geometric series is obtained by multiplying or dividing with the common number except the first term.

Q.2. What is the common ratio in the geometric progression?
Ans: It is the constant number that is used to multiply or divide each term of the geometric progression, excluding the first term.

Q.3. Is the reciprocal of the term of the given geometric series forms G.P.?
Ans: Yes. The reciprocal of the terms of the given geometric progression forms G.P.

Q.4. Is the series of numbers obtained by multiplying with constant, of the geometric progression forms G.P.?
Ans: Yes. The series of the numbers obtained by multiplying with a constant forms G.P.

Q.5. What is the general form of G.P.?
Ans: The general form of G.P. is \(a, a r, a r^{2}, a r^{3}, \ldots \ldots a r^{n}\)
Here, \(a\)-first term and \(r\)-common ration.

We hope this detailed article on the properties of geometric progression helped you in your studies. If you have any doubts, queries, or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!